Position multiple vertex to an exact location without changing shape of object
Based on the comments i build the setup myself, created the "P" face as it was called in place, took it down flat according to comments again and then started working it back in place.

Here a visual step by step:


It's based on the following:


Begin by creating a support triangle.
For that use K knife tool and hold Ctrl to snap on the middle of the base triangle edge that lies opposite of the tip where the "P" face has to go.
Select the top 2 vertices from each of the "green" standing triangles and press F to generate an edge between them.
Select the edge and press Ctrl+R to cut the edge in half, confirm with Enter
Now create a triangle from the new created vertex and the 2 vertices along the cut you made through the triangle. This will help aligning the "P" face.
Now go to the "P" face and in edit mode, then press Shift+S to snap the cursor to the selected vertex on the thin "P" face tip.
Go in object mode and set the origin to the cursor using the context-menu.
Switch to the other object and select the vertex where the thin tip of the "P" face has to go, snap the cursor there.
Back to the "P" face object use Shift+S "snap selected to cursor"
The "P" face object is now at least with the tip at the right location.
Adjust the settings for the snapping like so:

Now it's important that all vertices of the "P" face object are selected, BUT the last selected(active) vertex has to be the one at the tip opposite the thin tip where the origin is.

Press R, followed by Z, locking the rotation to Z axis and then hold Ctrl to snap based on the active vertex onto the vertex the support triangle, where it divides the base triangle edge.
Now back to the base object, in edit mode select one of the 2 edges of the divided edge on the base triangle opposite to the thin tip of the "P" face.
With the edge selected create a custom orientation like this:

It will create and select a custom orientation we need to rotate the "P" face.
Back to the "P" face object, select again all vertices of and make sure the last/active vertex is the one like we used last. Opposite the thin side.
Now press R, followed by Y (is the axis of the custom orientation we need), then hold Ctrl and snap based on the active vertex onto the top vertex of the support triangle.
All that is now needed is cleaning up the support triangle.
If you have multiple "P" faces that need to be placed, you can create a duplicate it and after removing the custom orientation snap each copy to the 2 other corners of the base object, after which you only need to rotate them with R, followed by Z with 120 degree either plus or minus.


With one extra "P" face it should then look like this:


I have looked at the file, and i had to notice that the "P" faces were slightly too long...

Little extra, as i was already mostly done with the answer:
To round it up i think it might help showing you how i build the version i used:


Add a circle mesh with 3 vertices, radius 1
Select an edge and create a custom orientation from it.
Snap the cursor to the edge
Select the vertex opposite the edge, copy it with Shift+D, cancel with RMB to leave it where it was.
Press R, then Y followed by -90 to position the vertex where it needs to be to form a new triangle you can now build by selecting the new vertex and the two on each side of the edge in between which the cursor is.
Do the same on the other side and you have the 2 "green" triangles standing.
Now select the 2 vertices each top of the "green" vertices and press F to create an edge between them.
Cut the edge with Ctrl+R, confirmed with LMB.
Select that new vertex and the vertex in the corner where the "P" face thin side will be and press F, creating an edge we can use to find the opposite tip of the "P" face.
Looking from top, select the vertex between the tips of the 2 "green" triangles and press G twice to start sliding the vertex along the long edge downwards.
While in this slide mode, press C to disable the clamping.
Now move along the long edge and over the cross-section towards the center of the base triangle. We can sadly not snap in this mode yet, but you will find that the closer to 0 on x and y you come the more you come closer to 2 on Z.
That said, with the values used the exact position would be 0,0,2.
Now with the vertex in place, you can create a face on each side and then dissolve the edge in the middle.


Based on what values you used i believe it should have come to fit, but since the version in the file was overlapping, either moving the vertices caused that or some measurements could have been slightly off.

Happy Blending.